calculus 7.5-7.9. 7.5 indeterminant forms l’hopital’s rule if f(a)=g(a)=0,
TRANSCRIPT
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Calculus
7.5-7.9
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7.5 Indeterminant Forms
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L’Hopital’s Rule
If f(a)=g(a)=0,
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L’Hopital’s Rule
If f(a)=g(a)=0,
f’(a), g’(a) exist, g’(a) = 0 NOT
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L’Hopital’s Rule
If f(a)=g(a)=0,
f’(a), g’(a) exist, g’(a) = 0 NOT,
then lim x a f(x) = f’(a)
g(x) g’(a)
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Examples
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Other indeterminant forms are
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Examples
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7.6 Rates at which functions grow
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f grows faster than gas x approaches infinity if
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f and g grow at the same rate as x approaches infinity if
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example
Show y=e^x grows faster than y= x^2
as x approaches infinity.
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example
Show y= ln x grows more slowly than y=x as x approaches infinity.
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example
Compare the growth of y=2x and y=x as x approaches infinity.
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7.7 trig review
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This is a picnic !!!!!
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7.8 derivatives of inverse trig functions
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7.8 integrals of inverse trig functions
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7.9 Hyperbolic Functions
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Def of hyperbolic functions
cosh x =
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Def of hyperbolic functions
cosh x =
sinh x =
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Def of hyperbolic functions
cosh x =
sinh x =
tanh x =
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Def of hyperbolic functions
cosh x =
sinh x =
tanh x =
sech x =
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Def of hyperbolic functions
cosh x =
sinh x =
tanh x =
sech x =
csch x =
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Def of hyperbolic functions
cosh x =
sinh x =
tanh x =
sech x =
csch x =
coth x =
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Identities
cosh^2 – sinh^2 = 1
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Identities
cosh^2 x– sinh^2 x= 1
cosh 2x = cosh^2 x + sinh^2 x
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Identities
cosh^2 x – sinh^2 x = 1
cosh 2x = cosh^2 x + sinh^2 x
sinh 2x = 2 sinh x cosh x
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Identities
cosh^2 x – sinh^2 x = 1
cosh 2x = cosh^2x + sinh^2x
sinh 2x = 2 sinh x cosh x
coth^2 x = 1 + csch^ 2 x
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Identities
cosh^2 x – sinh^2 x = 1
cosh 2x = cosh^2x + sinh^2x
sinh 2x = 2 sinh x cosh x
coth^2 x = 1 + csch^ 2 x
tanh^2 x = 1- sech^2 x
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These are cool
cosh 4x + sinh 4x =
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clearly
cosh 4x – sinh 4x =
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therefore
sinh e^(nx) +
cosh e^(nx) = e^(nx)
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(sinh x + cosh x ) = e^x
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(sinh x + cosh x ) = e^x
So
( sinh x + cosh x )^4 = (e^x)^4
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(sinh x + cosh x ) = e^x
So
( sinh x + cosh x )^4 = (e^x)^4
= e^(4x)
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MORE
sinh (-x) = - sinh x
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MORE
sinh (-x) = - sinh x
cosh (-x) = cosh x
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Derivatives of hyperbolic functions
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Integrals of hyperbolic functions
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Can you guess what’s next?
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Of course!
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Inverse hyperbolic functions
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Inverse hyperbolic functions
Derivatives
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Inverse hyperbolic functions
Integrals
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7.5 – 7.9 Test